SOLUTION: Simplify: 1 - csc^2 (theta)/cot^2 (theta)

Algebra ->  Trigonometry-basics -> SOLUTION: Simplify: 1 - csc^2 (theta)/cot^2 (theta)      Log On


   

Question 349080: Simplify: 1 - csc^2 (theta)/cot^2 (theta)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
(Algebra.com's software will not do theta. So I will use "x" instead.)
%281+-+csc%5E2%28x%29%29%2Fcot%5E2%28x%29

When trying to manipulate and simplify expressions with trig functions, it is often helpful to rewrite the expression using only sin and/or cos. The other 4 functions can be written in terms of sin/cos. Using this idea on your expression, and using the facts that csc = 1/sin and cot = cos/sin, we get:
%281+-+%281%2Fsin%5E2%28x%29%29%29%2F%28cos%5E2%28x%29%2Fsin%5E2%28x%29%29
With sin%5E2%28x%29 as the denominator of both "little" fractions, we can simplify this expression by multiplying the numerator and denominator of the "big" fraction by sin%5E2%28x%29:

On top we will need to use the Distributive Property to multiply:
%28sin%5E2%28x%29+-+1%29%2Fcos%5E2%28x%29
We know that cos%5E2%28x%29+=+1+-+sin%5E2%28x%29. So what is sin%5E2%28x%29+-+1? Answer: -cos%5E2%28x%29!! Substituting this into the expression we get:
%28-cos%5E2%28x%29%29%2Fcos%5E2%28x%29
The cosines cancel leaving
%28-1%29%2F1+=+-1
Your expression simplifies down to -1!